Who this is for: Cambridge IGCSE Mathematics (0580/0607) students who want Trigonometric Equations — solving sin x = k, cos x = k and tan x = k within a given range — to become a reliable source of marks instead of a topic they only half-remember.
What query it owns: how to understand and revise Trigonometric Equations in Cambridge IGCSE Mathematics.
Why this is safe: this page owns the Trigonometric Equations revision-guide angle, while Tutopiya’s Trigonometric Equations subtopic page owns the learning resource and the free Trigonometric Equations quiz owns the practice.

Trigonometric Equations appear in the Trigonometry unit of Cambridge IGCSE Mathematics (0580/0607) whenever examiners ask you to find all angles that satisfy an equation such as 2 sin x = 1 or cos x = −0.5 within a stated interval. The method combines inverse trig on your calculator with symmetry from the unit circle. This guide explains exactly what the subtopic covers, how to handle the question types that actually appear, and where to practise each skill.

Key takeaways

• A trigonometric equation asks for all angles x that make an expression like sin x, cos x or tan x equal a given value.
• Use sin⁻¹, cos⁻¹ or tan⁻¹ on your calculator for the principal value, then find other solutions using symmetry.
• Always check the stated range — e.g. 0° ≤ x ≤ 360° or 0 ≤ x ≤ 2π — and list every valid answer.
• Sketch or recall the trig graph to spot the second (and third) solution quickly.

What are Trigonometric Equations in Cambridge IGCSE Maths?

Trigonometric Equations are equations involving sine, cosine or tangent where the unknown is the angle. In Cambridge IGCSE Mathematics you solve equations such as sin x = 0.6 or 3 cos x + 1 = 0 by rearranging to standard form, using inverse trig for one solution, then applying graph symmetry to find every angle in the given range. Extended papers may include equations like 2 sin x = cos x that require dividing by cos x or using identities.

You can read the full explanation, worked examples and notes on Tutopiya’s Trigonometric Equations subtopic page before you attempt questions.

The core ideas you must master

These four ideas appear again and again. Learn what each one means and the exam phrasing that signals it.

Idea	What it means	How the exam uses it
Standard form	Isolate sin x, cos x or tan x on one side	”Solve 2 sin x − 1 = 0” → sin x = 0.5
Principal value	First answer from sin⁻¹, cos⁻¹ or tan⁻¹	Calculator gives one angle in the correct quadrant
Symmetry	Second solution from graph reflection	sin: 180° − θ; cos: 360° − θ; tan: add 180°
Stated range	Only list angles inside the interval given	”0° ≤ x ≤ 360°” — do not give x = 420°

How to solve a trigonometric equation — step by step

The safest method works for every equation of the form trig(x) = k.

1. Rearrange so sin x, cos x or tan x is alone on one side.
2. Use inverse trig on your calculator for the principal value (check DEG/RAD mode matches the question).
3. Find other solutions using symmetry: for sin x = k in 0°–360°, the second answer is 180° − θ; for cos x = k, use 360° − θ; for tan x = k, add 180°.
4. List every answer within the stated range — there may be two, three or four solutions.
5. Check by substituting each value back into the original equation.

Once you have worked through a few, test yourself with the free Trigonometric Equations quiz — it tells you fast whether the method has actually stuck.

sin x vs cos x vs tan x: which symmetry rule applies?

Students lose marks by applying the wrong reflection rule or forgetting that tan repeats every 180°. Use the equation type to decide.

Equation type	Symmetry for second solution (degrees)	Typical signal words
sin x = k	180° − θ	”Solve sin x = …“
cos x = k	360° − θ	”Solve cos x = …“
tan x = k	θ + 180°, θ + 360°, …	”Solve tan x = …”
Mixed (e.g. sin x = cos x)	Divide by cos x → tan x = 1, or use graph	”Solve 2 sin x = cos x”

Trigonometric Equations in past-paper wording: command words that matter

Most lost marks come from misreading the command word or giving only one solution when the range allows two. These are the phrasings you will see.

Command word / phrase	What the question wants	Typical stem
Solve	Find every value of x in the range, showing working	”Solve the equation 2 sin x = 1 for 0° ≤ x ≤ 360°.”
Write down	State answer(s); may be 1 mark each	”Write down the two values of x.”
Give your answers correct to 1 decimal place	Round as instructed	”Give your answers correct to 1 decimal place.”
Show that	Prove a given result — method earns marks	”Show that x = 30° is a solution of sin x = 0.5.”
Find all the values of x	Every solution in range — not just one	”Find all the values of x for which cos x = −0.5.”

Worked exam-style stems (how to answer the wording)

Practising the wording — not just the formula — is what method marks reward.

1. “Solve sin x = 0.5 for 0° ≤ x ≤ 360°.” x = sin⁻¹(0.5) = 30°. Second solution: 180° − 30° = 150°. Mark-scheme reward: both angles listed.
2. “Solve 3 cos x + 1 = 0 for 0° ≤ x ≤ 360°.” cos x = −⅓ → x ≈ 109.5° or 360° − 109.5° = 250.5°. Reward: rearrangement shown before inverse trig.
3. “Solve tan x = 1 for 0° ≤ x ≤ 360°.” x = 45° and 45° + 180° = 225°. Reward: both values from the 180° period of tan.

When you can recognise the wording instantly, work the full set on the Trigonometry topical past-paper questions and the Trigonometric Equations quiz to lock the method in.

How Trigonometric Equations connect to the rest of Trigonometry

Trigonometric Equations build directly on Trigonometric Graphs, where you learn the shape and period of sin, cos and tan — the graphs tell you how many solutions exist in a range. They also follow Right Angled Trigonometry and link forward to 3D Trigonometry. When you are ready to mix topics, the Cambridge IGCSE Maths resource hub lets you move straight from a weak subtopic into the next.

Common mistakes students make

• Giving only one solution when the range 0°–360° allows two (or more for tan).
• Using 180° − θ for cos x = k instead of 360° − θ.
• Forgetting to check DEG vs RAD mode on the calculator.
• Rounding the principal value too early and losing accuracy on the second solution.
• Listing angles outside the stated range — e.g. giving x = 390° when the range ends at 360°.

When you need more support

If trigonometric equations keep tripping you up — especially finding the second solution — work through the Trigonometry topical past-paper questions and the Trigonometric Equations quiz to pinpoint the exact gap, then get focused help from a Cambridge IGCSE Maths tutor to fix it quickly.

Frequently asked questions

Are Trigonometric Equations hard in Cambridge IGCSE Maths? The method is straightforward once you know the symmetry rules. Marks are lost when students give only one solution or use the wrong reflection for sin vs cos.

How do I find the second solution for sin x = k? Calculate θ = sin⁻¹(k), then the second answer in 0°–360° is 180° − θ. Always check both satisfy the original equation.

Do I need to know radians for trigonometric equations? Some papers use 0 ≤ x ≤ 2π instead of degrees. The same symmetry rules apply — just work in radians throughout and keep your calculator in RAD mode.

How do I revise Trigonometric Equations effectively? Read the subtopic notes, sketch the relevant trig graph for each equation type, then take the Trigonometric Equations quiz. Revisit any questions where you missed a second solution.

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